Digital Signal Processing applied to Physical Signals

POLITECNICO DI TORINO SCUOLA DI DOTTORATO Dottorato in Ingegneria Elettronica e delle Comunicazioni

Tesi di Dottorato

Digital Signal Processing applied to Physical Signals

Diego ALBERTO

Tutore Prof. Roberto Garello

Coordinatore del corso di dottorato Prof. Ivo Montrosset

Marzo 2011

If you can talk with crowds and keep your virtue, Or walk with kings - nor lose the common touch; If neither foes nor loving friends can hurt you; If all men count with you, but none too much; If you can fill the unforgiving minute With sixty seconds’ worth of distance run Yours is the Earth and everything that’s in it, And - which is more - you’ll be a Man my son! Rudyard Kipling - If

Summary It is well known that many of the scientific and technological discoveries of the XXI century will depend on the capability of processing and understanding a huge quantity of data. With the advent of the digital era, a fully digital and automated treatment can be designed and performed. From data mining to data compression, from signal elaboration to noise reduction, a processing is essential to manage and enhance features of interest after every data acquisition (DAQ) session. In the near future, science will go towards interdisciplinary research. In this work there will be given an example of the application of signal processing to different fields of Physics from nuclear particle detectors to biomedical examinations. In Chapter 1 a brief description of the collaborations that allowed this thesis is given, together with a list of the publications co-produced by the author in these three years. The most important notations, definitions and acronyms used in the work are also provided. In Chapter 2, the last results on the filter designs to be implemented in the trigger-less DAQ of the PANDA experiment are presented. Results obtained from simulation are used as basis for some FPGA-oriented filter projects able to process in real time data from nuclear particle detectors. For all studied filter structures, particular attention has been paid to the board inner-components consumption and to the maximum working frequency, since our aim is an on-line treatment. In Chapter 3, from a collaboration with the INRiM institute of research, the results of signal processing of data coming from TES single photon detectors are reported. Because of their high sensitivity and fast response, the application of these detectors is mandatory to all fields that deal with weak sources. It ranges from astrophysics to structure of matter, from X-rays to infra-red wavelengths. However, the electronic DAQ system and the environmental conditions of every acquisition require a digital treatment to extract the most important features of interest as the energy resolution or the capability to really count single photons of every DAQ session. The author had the possibility to spend a three months period as a PhD visiting student at CERN, Geneva. During this experience he was involved in the ALICE experiment of the LHC project. The goal was the development of an algorithm able

to process on-line data from all detectors to be implemented on FPGA. The results of this study are reported in Chapter 4, subdivided in a simulation part, used to understand and write the algorithm, and in a real data analysis. The last application of signal processing presented, comes from a collaboration with the Ottica e Optometria course of studies of the University of Turin, it is introduced in Chapter 5. The analyses conduced on human corneas aiming at distinguishing all corneal sub-layers and estimating their thicknesses are reported. Every acquired image represents a signal to be processed. The impact of this development on medical applications is very high since, for the first time, all these clinical tests can be made in-vivo with no adverse effect for patients and with a precision never reached before. The work is concluded, in Chapter 6, with some considerations on the usefulness, and surely the necessity, of signal processing in science experiments.

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Acknowledgments ... to (and notwithstanding) Andrea, Chiara and Arianna I would like to thank: • Prof. M.P.Bussa, Dr. A.Grasso, the Torino Panda-mu Group for their scientific support during this three years at Dipartimento di Fisica Generale of Università di Torino and, in particular, Dr. M.Greco, my Physics supervisor. • Prof. R.Garello my PhD supervisor at Dipartimento di Elettronica of Politecnico di Torino and a dear friend. • Dr. E.Falletti and Dr. F.Molinari for their advices in the DSP subject. • Prof. G.Masera, Dr. A.Dassatti and Dr. L.Toscano for the useful suggestions on FPGA implementations of filter structures. • Dr. M.Frisani for having provided the OCT corneal images and the time spent in many discussions on OCT acquisitions. • Dr. M.Rajteri, Dr. L.Lolli and the TES group at INRiM for the collaboration on TES pulse analyses. • Dr. L.Musa and Dr. P.G.Innocenti for the PhD visiting student period at CERN and for the possibility to cope with real data from ALICE experiment. • Dr. M.Destefanis for the discussions regarding Physics and LATEXthemes. • PhD students Alessandro Re, Fabrizio Sordello, Isacco Scanavino, Ivan Gnesi, Marco Musich and Thanushan Kugathasan for the pleasant lunchtime discussions on whatever known (or unknown) subject and for their friendship. Finally, a special thank to all my family for the precious help given to babysitting our little children and, in particular, to my wife Gabriella, for the way in which she trusts me. iv

Ringraziamenti (Italian Version) ... a (e nonostante) Andrea, Chiara e Arianna Vorrei ringraziare: • la Prof. M.P.Bussa, l’Ing. A.Grasso, il gruppo Panda-mu Torino per il loro supporto scientifico durante questi tre anni al Dipartimento di Fisica Generale dell’Università di Torino e, in particolare, la Dr. M.Greco, mia responsabile a Fisica. • Il Prof. R.Garello mio tutore di dottorato al Dipartimento di Elettronica del Politecnico di Torino e caro amico. • L’Ing. E.Falletti e l’Ing. F.Molinari per i loro consigli in materia di analisi dei segnali. • Il Prof. G.Masera, l’Ing. A.Dassatti e l’Ing. L.Toscano per gli utili suggerimenti sull’implementazione FPGA delle strutture di filtraggio. • Il Dr. M.Frisani per aver fornito le immagini delle cornee analizzate e per il tempo speso in molte discussioni sulle acquisizioni OCT. • L’Ing. M.Rajteri, l’Ing. L.Lolli ed il gruppo TES all’INRiM per la collaborazione all’analisi degli impulsi dai rivelatori TES. • Il Dr. L.Musa e l’Ing. P.G.Innocenti per il periodo di visita al CERN in qualità di studente di dottorato e per la possibilità di confrontarmi con i dati reali dell’esperimento ALICE. • Il Dr. M.Destefanis per le discussioni riguardo tematiche come fisica e LATEX. • I dottorandi Alessandro Re, Fabrizio Sordello, Isacco Scanavino, Ivan Gnesi, Marco Musich e Thanushan Kugathasan per le simpatiche discussioni su qualsiasi materia nota (o ignota) all’ora di pranzo e per la loro amicizia. Infine uno speciale ringraziamento va a tutta la mia famiglia per il prezioso aiuto fornitoci nella gestione dei bimbi e, in particolare, a mia moglie Gabriella, per il modo in cui confida in me. v

Table of contents Summary

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Acknowledgments

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1 Introduction 1.1 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Scientific publication . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Notations, Definitions and Acronyms . . . . . . . . . . . . . . . . . . 2 Nuclear particle signals analysis for PANDA experiment 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 System model and simulation scheme . . . . . . . . . . . . . 2.3 Digital conversion and filtering algorithms . . . . . . . . . . 2.3.1 Butterworth LP digital filter . . . . . . . . . . . . . . 2.3.2 MMSE noise canceller . . . . . . . . . . . . . . . . . 2.4 Behaviour comparison of simulated results . . . . . . . . . . 2.5 FPGA-oriented implementations . . . . . . . . . . . . . . . . 2.5.1 Butterworth LP digital filter . . . . . . . . . . . . . . 2.5.2 MMSE noise canceller . . . . . . . . . . . . . . . . . 2.6 Discussion of FPGA-oriented results . . . . . . . . . . . . . 2.6.1 Filter performance . . . . . . . . . . . . . . . . . . . 2.6.2 Power consumption . . . . . . . . . . . . . . . . . . . 2.6.3 Maximum working frequency . . . . . . . . . . . . . 2.7 A statistical approach . . . . . . . . . . . . . . . . . . . . . 2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Transition-Edge Sensor single 3.1 Introduction . . . . . . . . . 3.2 Experimental Setup . . . . . 3.3 Signal Analysis . . . . . . . 3.3.1 Energy Resolution .

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pulse analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.4 3.5 3.6 3.7

3.3.2 Signal-to-Noise Ratio . . . . . . . . . . . 3.3.3 Time Jitter . . . . . . . . . . . . . . . . 3.3.4 Savitzky-Golay filter . . . . . . . . . . . 3.3.5 Wiener filter . . . . . . . . . . . . . . . . Results and discussion . . . . . . . . . . . . . . Other examples of analysed datasets . . . . . . Energy evaluation and TES amplitude response Conclusions and outlook . . . . . . . . . . . . .

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4 Amplitude estimation of real signals for the 4.1 Introduction . . . . . . . . . . . . . . . . . . 4.2 ALICE experiment . . . . . . . . . . . . . . 4.3 Self-adaptive piecewise linear filter . . . . . 4.3.1 Simulation results: ideal case . . . . 4.3.2 Simulation results: ideal case + noise 4.4 TPC real-data analysis . . . . . . . . . . . . 4.5 Conclusions . . . . . . . . . . . . . . . . . .

ALICE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Digital processing of OCT corneal images 5.1 Introduction . . . . . . . . . . . . . . . . . 5.2 Experimental setup . . . . . . . . . . . . . 5.3 Our purpose . . . . . . . . . . . . . . . . . 5.4 Our approach . . . . . . . . . . . . . . . . 5.5 OCT images and SNR . . . . . . . . . . . 5.6 Treatment of corneal marginal regions . . 5.7 Conclusions . . . . . . . . . . . . . . . . .

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6 Final conclusions

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A The sign-LMS adaptive filter

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B The uncertainty on TES Energy Resolution

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C The averaging procedure and SNR improvement

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D Wiener filter and SNR

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E The Buzuloiu’s algorithm for 2D signals

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F Simulation of a 2D corneal layer reconstruction

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Bibliography

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Chapter 1 Introduction 1.1

Thesis outline

This PhD thesis is the fruit of a collaboration between: Dipartimento di Elettronica of Politecnico di Torino and Università degli Studi di Torino, Dipartimento di Fisica Generale. The main part of this work has been devoted to the development of digital filters for the PANDA nuclear particle experiment thanks to a collaboration between: • the P ANDA group and the Istituto Nazionale di Fisica Nucleare (INFN) Turin section. Will also be presented the analysis of data coming form collaborations between Dipartimento di Fisica Generale and: • Istituto Nazionale di Ricerca Metrologica (INRiM) for the elaboration of single photon pulses. • Corso di Studi in Ottica e Optometria for the analysis of human corneal images. • European Organization for Nuclear Research (CERN) for the processing of real data form ALICE experiment.

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1 – Introduction

This thesis deals with the digital treatment of signals coming from different fields of Physics, as reported in the Summary Section. Digital Signal Processing (DSP) can be applied whenever a signal is digitalized to: • reduce noisy (unwanted) components; • enhance desired ones; • extract features of interest (amplitude, peak time, signal energy, ..); • estimate indirect measurements (particle energy, energy resolution, ..); • process on-line real data (with FPGA); • process off-line measured and saved waveforms. All analyses presented in the following Sections have been performed by means of Matlab programs [1], Simulink block diagrams [1], and Xilinx tools [2] for the VHDL implementation of filter structures on FPGA devices.

1.2

Scientific publication

This introductory Chapter continues with a list of articles and presentations where some of the results here described have been published.

International Journals : • Optical transition-edge sensors single photon pulses analysis, D.Alberto, M.Rajteri, E.Taralli, L.Lolli, C.Portesi, E.Monticone, Y. Jia, R.Garello, M.Greco, to be published on IEEE Transaction on Applied Superconductivity, June 2011, DOI: 10.1109/TASC.2010.2087736. • Ti/Au Transition-Edge Sensors Coupled to Single Mode Optical Fibers Aligned by Si V-Groove, L.Lolli, E.Taralli, C.Portesi, D.Alberto, M.Rajteri, E.Monticone, to be published on IEEE Transaction on Applied Superconductivity, June 2011. • FPGA implementation of digital filters for nuclear detectors, D.Alberto, E.Falletti, L.Ferrero, R.Garello, M.Greco, M.Maggiora, NUCLEAR INST. AND METHODS IN PHYSICS RESEARCH, A, 611: 99-104 October 2009, DOI: 10.1016/j.nima.2009.09.049. 2

1 – Introduction

• Digital Filtering for Noise Reduction in Nuclear Detectors, D. Alberto, M.P.Bussa, E.Falletti, L.Ferrero, R.Garello, A.Grasso, M.Greco, M.Maggiora, NUCLEAR INST. AND METHODS IN PHYSICS RESEARCH, A, 594 (3): 382-388 September 2008, DOI: 0.1016/j.nima.2008. 06.032 -2008. • Effects of Extremely Low-Frequency Magnetic Fields on L- glutamic Acid Aqueous Solutions at 20, 40, and 60 µT Static Magnetic Fields, D. Alberto, L.Busso, R.Garfagnini, P.Giudici, I.Gnesi, F.Manta, G.Piragino, L.Callegaro, G.Crotti, ELECTROMAGNETIC BIOLOGY AND MEDICINE, pp. 241-253, 2008, Vol. 27, ISSN: 1536-8378, DOI: 10.1080/15368370802344052. • Effects of Static and Low-Frequency Alternating Magnetic Fields on the Ionic Electrolytic Currents of Glutamic Acid Aqueous Solutions, D. Alberto, L.Busso, G.Crotti, M.Gandini, R.Garfagnini, P.Giudici, I.Gnesi, F.Manta, G.Piragino, ELECTROMAGNETIC BIOLOGY AND MEDICINE, pp. 25-39, 2008, Vol. 27, ISSN: 1536-8378, DOI: 10.1080/15368370701878788. International Conferences : • Ti/Au Transition-Edge Sensors Coupled to Single Mode Optical Fibers Aligned by Si V-Groove, L.Lolli, E.Taralli, C.Portesi, D.Alberto, M.Rajteri, E.Monticone, Y. Jia, IEEE Applied Superconductivity Conference - Washington D.C., 1-6 August 2010. • Optical transition-edge sensors single photon pulses analysis, D. Alberto, M.Rajteri, E.Taralli, L.Lolli, C.Portesi, E.Monticone, Y.Jia, R.Garello, M.Greco, IEEE Applied Superconductivity Conference - Washington D.C., 1-6 August 2010. • Epithelial, Bowman’s layer, Stroma and pachimetry changes with FDOCT during orthokeratology, M.Frisani, D.Alberto, A.Calossi, M.Greco, 1st EuCornea Congress - Venice, 17-19 June 2010. • Digital Filtering of Particle Detector Signals, D.Alberto, M.Greco, M.Maggiora, S.Spataro, 17th IEEE Real Time Conference - Lisboa, 24-28 May 2010. • Epithelial, Bowman’s layer, Stromal and corneal thickness changes during orthokeratology by SD-OCT, M.Frisani, D.Alberto, M. Greco, EAOO - European Academy of Optometry and Optics 2010 Copenhagen, 15-16 May 2010. 3

1 – Introduction

• Digital Filters for Noise Reduction in Nuclear Detectors, D. Alberto, M.P.Bussa, E.Falletti, L.Ferrero, R.Garello, A.Grasso, M.Greco, M.Maggiora, 16th IEEE Real Time Conference - Beijing, 10-15 May 2009. National Conferences : • Filtraggio adattativo su segnali da rivelatori di particelle, XCVI Congresso Nazionale Società Italiana di Fisica (S.I.F.) - Bologna, 2024/09/2010. • La tomografia a coerenza ottica FD-OCT per lo studio morfometrico delle diverse componenti della cornea, M.Frisani, D. Alberto, M.Greco, A.Calossi, INOA - Istituto Nazionale Ottica Applicata CNR, Vinci, 7-8 November 2009. • Implementazione su FPGA di filtri digitali standard e adattativi per il trattamento di segnali da rivelatori di particelle, XCV Congresso Nazionale Società Italiana di Fisica (S.I.F.) - Bari, 28/09 03/10/2009. Winner of the first prize as the best presentation in section: Sezione V a - Fisica applicata (Sect. V a -Applied Physics). Online English version, Digital filters for nuclear particle detectors, D.Alberto, M.P.Bussa, E.Falletti, R.Garello, M.Greco, IL NUOVO CIMENTO B - Basic topics: Special Issue, 125 B (5-6), p. 677-686 June 2010, DOI:10.1393/ncb/i2010-10860-0. • Comparazione tra filtraggio digitale standard ed adattativo per la riduzione del rumore nei rivelatori di particelle, XCIV Congresso Nazionale Società Italiana di Fisica (S.I.F.) - Genova, 24/09/2008. • Filtraggio Digitale di Segnali Generati in Rivelatori di Particelle Nucleari, XCIII Congresso Nazionale Società Italiana di Fisica (S.I.F.) - Pisa, 24-29/09/2007. PANDA collaboration meetings : • Signal Processing for Nuclear Detectors, Panda DAQT Meeting Bavarian Forest, 22-24/04/2009. • Signal Processing on FPGA, Panda DAQT Meeting - Darmstadt, 8-10/12/2008. • Improvements in Digital Filtering for Nuclear Detectors, Panda DAQT Meeting - Torino, 18-19/06/2008. • Digital Filtering in Nuclear Detectors, Panda DAQT Meeting Juelich, 18-19/03/2008.

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1 – Introduction

1.3

Notations, Definitions and Acronyms

This Chapter ends with the most important notations, definitions and recurrent acronyms used in this work. Antiproton: the antiparticle of the proton. CP symmetry: it states that the laws of physics should be the same if a particle were interchanged with its antiparticle (C - charge conjugation - symmetry), and left and right were swapped (P - parity - symmetry). CP violation: a violation of the postulated CP symmetry. electron Volt (eV): the energy acquired by an electron when accelerated by a electric potential difference of 1 Volt and is equal to 1.602 · 10−19 C. Ergodic process: a stochastic process with statistical properties (such as its mean and variance) that can be deduced from a single, sufficiently long sample (extraction) of the process. Exotic State: a state of matter not foreseen by usual QCD calculations that includes quark and gluons (as hybrid hadrons) or only gluons (as glueballs). Filter: whatever block able to receive an input signal and produce an output waveform, even not modified (in this case it could be a simple delayer). • Standard F.: a filter with constant transfer function coefficients, once they have been calculated they do not change during the elaboration. • Adaptive F.: a filter that self-adjusts its transfer function according to an optimizing algorithm. Force: in physics, a quantitative description of the interaction between two physical bodies and can be of four fundamental types: • Gravitational: the force of attraction between all masses in the universe.

• Electro-Magnetic: associated with electric and magnetic fields and is responsible for atomic structure, chemical reactions, the attractive and repulsive forces associated with electrical charge and magnetism. • Weak: the fundamental force that acts between leptons and is involved in the decay of hadrons, it is also responsible for nuclear beta decay. • Strong: it mediates interactions between quarks and gluons. 5

1 – Introduction

Fourier Transform: the operation that decomposes a signal into its constituent frequencies. Hadron: a composite particle made of quarks held together by the strong force (i.e. protons and neutrons). Hadrons are categorized into two families: baryons (made of three quarks), and mesons (made of one quark and one antiquark). Hyperon: any baryon containing one or more strange quarks, but no charm or bottom quarks. Gluon: the particle that mediates the strong force. Lepton: an elementary particle and a fundamental constituent of matter, the best known of all leptons is the electron. Nucleon: a collective name for two particles: the neutron and the proton, these are the two constituents of the atomic nucleus. Orthokeratology: an ophthalmological treatment with contact lens that modifies the cornea’s shape changing its refraction properties. Phonon: in physics, a quasiparticle representing the quantization of the modes of lattice vibrations of periodic, elastic crystal structures of solids. Quark: an elementary particle and a fundamental constituent of matter, it can only be found within hadrons. There are six types of quarks, known as flavors: up, down, charm, strange, top, and bottom. Standard Model: a physical theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Stochastic process: a non-deterministic process where there is some indeterminacy in its future evolution described by probability distributions. Synchrotron: a particular type of cyclic particle accelerator in which the magnetic field (to turn the particles so they circulate) and the electric field (to accelerate the particles) are carefully synchronised with the travelling particle beam. WSS process: is a (Wide Sense Stationary) stochastic process whose joint probability distribution does not change when shifted in time or space. As a result, parameters such as the mean and variance, if they exist, also do not change over time or position.

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1 – Introduction

Here is provided a list of the acronyms that can be found throughout the text. ADC: ALICE: ALTRO: ASIC: ATLAS: CERN: CM: CMS: DAQ: DF: DFT: DGT: DSP: ER FAIR: FD: FEC: FIR: FPGA: FWHM: GSI: HESR: IIR: INRiM: IROC: ISR: KVL: LEP: LHC: LHCb: LINAC: LMS: LP: LSB: LUT: MMSE: MSE: MSPS:

Analog to Digital Converter A Large Ion Collider Experiment ALice Tpc Read Out chip Application Specific Integrated Circuits A Toroidal LHC ApparatuS Conseil Europeene pour la Recherche Nucleaire (European Organization for Nuclear Research) Center of Mass Compact Muon Solenoid Data AcQuisition Direct Form Discrete Fourier Transform Digital Digital Signal Processing Energy Resolution Facility for Antiproton and Ion Research Fourier Domain Front-End Card Finite Impulse Response Field Programmable Gate Array Full Width at Half Maximum Gesellschaft f¨ ur Schwerionenforschung High Energy Storage Ring Infinite Impulse Response Istituto Nazionale di Ricerca Metrologica Inner Read-Out Chamber Intersecting Storage Rings Kirchhoff’s Voltage Law Large Electron-Positron Collider Large Hadron Collider LHC beauty experiment LINear ACcelerator Least Mean Square Low Pass Least Significant Bit Look-Up Table Minimum MSE Mean Square Error Mega Samples Per Second 7

1 – Introduction

MWPC: OCT: PANDA: PASA: PD: PSD: QCD: QGP: RCU: RMS: SG: SNR: SPS: SQUID: TD: TES: TPC: VHDL: VHSIC: WGN: WSS:

Multi-Wire Proportional Chambers Optical Coherence Tomography anti-Proton ANnihilation at DArmstadt Pre-Amplifier Shaping Amplifier Peak Distortion Power Spectral Density Quantum Cromo-Dynamics Quark Gluon Plasma Readout Control Unit Root Mean Square Savitzky-Golay Signal-to-Noise Ratio Super Proton Synchrotron Superconducting Quantum Interference Device Time Domain Transition Edge Sensor Time Projection Chamber VHSIC Hardware Description Language Very High Speed Integrated Circuits White Gaussian Noise Wide Sense Stationary

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Chapter 2 Nuclear particle signals analysis for PANDA experiment 2.1

Introduction

The PANDA (anti-Proton ANnihilation at DArmstadt) Experiment will be one of the key experiments at the Facility for Antiproton and Ion Research (FAIR) which is currently being built on the area of the GSI Helmholtzzentrum f¨ ur Schwerionenforschung in Darmstadt (Fig.2.1), Germany [3], [4]. FAIR is an extension of the existing Heavy Ion Research Lab (GSI) and is expected to start its operation in 2019. The antiproton project was initiated by a large community of scientists outside GSI, who had worked very successfully with antiprotons at LEAR/CERN and at the Fermilab antiproton accumulator. Many of the physics ideas of PANDA were already described in a Letter of Intent (Construction of a GLUE/CHARM-Factory at GSI, Ruhr-University Bochum, 1999) and were extended afterwards in the FAIR Conceptual Design Report (GSI, 2001), the Technical Progress Report (FAIR, 2005) and further PANDA specific reports. After the approval of FAIR, further projects involving antiprotons were proposed (experiments with low energy and polarized antiprotons) which are now in the preparatory phase. The proposed project FAIR is an international accelerator facility of the next generation. It builds on the experience and technological developments already made at the existing GSI facility, and incorporates new technological concepts. At its heart is a double ring facility with a circumference of 1100 meters. A system of coolerstorage rings for effective beam cooling at high energies and various experimental halls will be connected to the facility. The existing GSI accelerators, together with the planned proton-linac, serve as injector for the new facility. The double-ring synchrotron will provide ion beams of 9

2 – Nuclear particle signals analysis for PANDA experiment

Figure 2.1: FAIR facility overview: at present (blue) and in the near future (red).

unprecedented intensities as well as of considerably increased energy. Thereby intense beams of secondary particles (unstable nuclei or antiprotons) can be produced. The system of storage-cooler rings allows the quality of these secondary beams to be drastically improved. Moreover, in connection with the double ring synchrotron, an efficient parallel operation of up to four scientific programs can be realized at a time. The central part of FAIR is a synchrotron complex providing intense pulsed ion beams (from proton to Uranium). Antiprotons produced by a primary proton beam will then be filled into the High Energy Storage Ring (HESR) which collide with the fixed target inside the PANDA Detector (Fig.2.2). The PANDA Collaboration with more than 450 scientists from 17 countries intends to do basic research on various topics around the weak and strong forces, exotic states of matter, the structure of hadrons, gluonic excitations, the physics of strange and charm quarks. In order to gather all the necessary information from the antiproton-proton collisions a versatile detector will be build able to provide precise trajectory reconstruction, energy and momentum measurements (in the range 1.5-15 GeV/c) and very efficient identification of charged particles. Protons and neutrons, collectively called nucleons, belong to the family of hadrons. They are built of quarks (see Fig. 2.3) and bound by the strong force that is mediated via gluons. The force acting between two quarks has an unusual behaviour: it is very small when the quarks are at close distance and increases as the distance grows. 10


Figure 2.2: PANDA spectrometer 3-dimensional view.

Figure 2.3: Quarks and their confinement pictorial representation.

If one attempts to separate a quark-antiquark pair, the energy of the gluon field becomes larger and larger. As a result, one does not end up with two isolated quarks but with new quark-antiquark pairs instead. This absolute imprisonment of quarks is called confinement. One of the greatest intellectual challenges of modern physics is to understand confinement not just as a phenomenon but to comprehend it quantitatively from the theory of the strong force. Another puzzle of hadron physics addresses the origin of the hadron masses, i.e. of the particles composed of quarks. In the nucleon, less than 2% of the mass can 11


be accounted for by the three valence quarks. Obviously, the bulk of the nucleon mass results from the kinetic energy and the interaction energy of the quarks confined in the nucleon. Physicists believe that new experiments exploiting high-energy antiproton and ion beams will also elucidate the generation of hadronic masses. The PANDA scientific program includes several measurements [5], which address fundamental questions of Quantum Cromo-Dynamics (QCD), mostly in the nonperturbative regime: • Hadron spectroscopy up to the region of charm quarks. Here the search for exotic states like glueballs, hybrids and multiquark states in the light quark domain and in the hidden and open charm region is in the focus of interest. • Study of properties of hadrons inside nuclear matter. Mass and width modifications have been reported and will be investigated also in the charm region. • Study of nonperturbative dynamics, also including spin degrees of freedom. • Antiproton induced reactions are a very effective tool to implant strange baryons in nuclei. • Hard exclusive antiproton-proton reactions can be used to study the structure of nucleons (time-like form factors) and the relevance of certain models, like the Hand Bag approach. Interesting aspects of Transverse Parton Distributions will be studied in Drell-Yan production. • In a later stage of the project, when all systematic effects are well studied, also contributions to electroweak physics can be expected, like direct CP violation in hyperon decays and CP violation and mixing in the charm sector. All measurements will profit from the high yield of antiproton induced reactions and from the fact that, in contrast to e+ e− reactions, all non-exotic quantum number combinations for directly formed states are allowed, whereas states with exotic quantum numbers can be observed in production. The achievable precision, as far mass and width measurements are concerned, is very high as was successfully demonstrated by the Fermilab experiments. The physics purpose of PANDA has been described above. However, the area in which our work will focuses is more engineering-oriented, it is the Data-Acquisition (DAQ) section of the experiment. The standard architecture of a DAQ systems in nuclear detectors is based on a two-layer hierarchical approach. A subset of especially instrumented detectors is used to evaluate a first-level trigger condition. For the accepted events, the full information of all detectors is then transported to the next higher trigger level or 12


to storage. The time available for the first-level decision is usually limited by the buffering capabilities of the front-end electronics [6] - [8]. The next generation of experiments in the hadron facilities, like the FAIR one at Darmstadt, will study rare events at a drastically improved sensitivity. Interesting signals will only become available by a combination of high interaction rates (normally higher than 10 MHz), fast detectors and broad bandwidth data acquisition systems to select in a fitting way only the events of interest. These constraints make it necessary to go beyond the old two-layer hierarchical approach towards selftrigger systems. They autonomously detect signals and pre-process them to extract and transmit only the physically relevant information for further processing. This means that they are able to discriminate how relevant the event is and, if required to select it, to select means to filter in the right way and dynamically the signals (see for example [9]). We aim to develop a data acquisition system through the study of specific algorithms for the reliable detection of informative pulses (i.e., the pulses generated by the interaction of a charged particle) partially buried in noise, as well as their implementation on electronic boards which use programmable devices. In the following Sections a system model for the DAQ and a set of standard and adaptive filters aiming to the noise reduction of noisy simulated signals [10] will be presented. These digital filters will be compared on their capability to extract features of interest from acquired pulses (amplitude, peak distortion, Signal to Noise ratio - SNR ). The translation of the Matlab-simulated filtering structures into hardware-oriented ones and their implementation on FPGA devices for the on-line processing [11] will also be presented. This Chapter will end with a statistical approach on the performed simulations [12].

13


2.2

System model and simulation scheme

When a charged particle is detected, the detector produces a current pulse that is processed through a transmission chain. This signal is affected by several causes of noise (thermal, shot, flicker, etc.) which impair the correct pulse detection. We aim to develop a filtering system able to reduce this noise as much as possible. The elaboration will be performed in the digital domain and for this reason the signal must be processed by an Analog to Digital Converter (ADC). However, because of the reduced input bandwidth (20 MHz) and the sampling rate of the ADCs currently available, the bandwidth of the input signal (i.e., informative pulses plus noise) has to be properly reduced. Thus, a Low-Pass (LP) analog transmission chain had to be introduced before the ADC. A possible model is presented in Fig. 2.4, together with a pictorial representation of the impulse response. The analog section is composed of the following: • Detector: it detects charged particles and produces informative pulses with amplitude proportional to the charge (Q). • Preamplifier/integrator: it integrates the input signal and at the same time reduces the bandwidth. When the current pulse is present, the integrator produces a signal proportional to the charge. The gain kp is used to normalize the peak to a fixed value; τp is the time constant of the preamplifier/integrator. • PoleZero compensator: it introduces a faster pole erasing the preamplifier one, in order to enlarge the total pass-band and thus to avoid as much as possible distortion of the informative pulses, which can cause a pile-up effect of the filtered pulses. The gain kpz is used to normalize the peak to a fixed value, while the time constant τpz τp . • Analog shaper/antialias filter: it represents the final LP antialiasing filter, opportunely matched to the ADC input bandwidth and sampling rate. In practice, it enlarges the top of the signal, allowing the ADC to obtain more than one significant sample for each informative pulse. The gain kAS is used to normalize the peak to a fixed value, tAS = 1/fB is the time constant, where fB is the desired signal band, and n is the denominator exponent. The factor n determines the slope of the transfer function in the transition bandwidth and the top flatness of the signal. However, the higher the parameter n is, the more difficult the hardware assemblies are. Hence, n was chosen as a compromise between the hardware complexity and the minimization of the sampling error.

14


Figure 2.4: Analog transmission sub-chain.

After these three blocks the shaped signal is ready to be converted by the digital sub-chain (Fig. 2.5): • ADC: it converts the analog signal into a digital one; • Noise filter: it is digital and is designed to possibly reduce the noise that affects the desired signal. It can be standard or adaptive, with Finite (FIR) or Infinite (IIR) Impulse Response. In general, an analog signal and an ADC sampler are not synchronous in time. In the analog-to-digital conversion, focussing on the sampling process, the analog maximum could not be sampled and a Peak Distortion (PD) could be introduced. This happens because the highest sampled value is considered the maximum and it could not correspond to the analog maximum. Here, with PD we intend the relative difference between the analog measured peak value and the sampled one (usually expressed in percentage). It is important to remember that in this subsection we are not considering the quantization, that is the process by which an analog quantity (i.e. a sampled value) is converted into a digital one (represented with a finite number of

Figure 2.5: Digital transmission sub-chain.

15


bits). The quantization error cannot be avoided but depends on the number of bits the ADC is capable, thus, choosing a more performant ADC, it can be reduced. As we can see in Fig. 2.6, different asynchronous samplings lead to different PD (∆Pi in the Figure). The worst situation arises when we have two samples with the same highest value (i.e. blue squares in Fig. 2.6). They correspond to the highest PD (∆P 1 ) because in any other case there is only one sample with the highest value, so the related distortion is lower (i.e. ∆P 2 in the same Figure). The ideal case is represented by a distortion equal to zero (i.e. ∆P 3 in Fig. 2.6) that can be obtained only when a casual synchronization between signal and sampler occurs. In our case the sample time TS is 10 ns (being the sampling frequency of 100 MHz) and represents the time distance between two consecutive samples.

Figure 2.6: Pictorial representation of different asynchronous samplings.

16


In order to obtain one sample as close as possible to the analog signal maximum, we have to enlarge the signal top. This operation is performed introducing the Analog Shaper block and paying particular attention to the choice of the n term, that is the Shaper (denominator) degree. Neglecting the hardware fabrication complexity, in Fig. 2.7 we have evaluated the PD (expressed as percentage error) as a function of the Shaper order n. This distortion is expressed in percentage with respect to the analog value and presents a decreasing behaviour because the higher the n is, the more flat the signals tops are, consequently the PDs are reduced. The Shaper’s time constant τAS has been set to 50 ns in according to a foreseen band of interest of 20 MHz (for this choice, see Sect. 2.3).

2.5

2

% error

1.5

1

0.5

0

0

2

4

6

8 10 shaper order

12

14

Figure 2.7: Peak Distortion as a function of the Shaper order n..

17

16


However the shaper order n should be chosen also considering the pile-up effect. A high n introduces a low PD but also a high top signal enlargement, this enlargement increases the separation time needed by our board to detect as separate two different consecutive particles. In Fig. 2.8 the minimum inter-arrival time between two particles, to be detected as different ones by our board, as a function of our shaper order n is shown. For any value of n, if two particles are separated in time less than the corresponding time value given by Fig. 2.8, a pile-up effect occurs. Also in this case the time constant τAS has been set to 50 ns in according to the value of the previous simulation. 1800

1600

inter. time [ns]

1400

1200

1000

800

600

400

0

2

4

6

8 10 shaper order

12

14

16

Figure 2.8: Inter-arrival time between two particles as a function of the Shaper order n.

In order to evaluate the improvement obtained using a digital filter to partially suppress the noise, in our simulation we modelled the information pulses with a series of successive finite support waveforms, with very narrow time duration (0.5 ns) and random times of arrival. No specific assumptions have been made at this stage about the mathematical model of the times of arrival, since experimental results are not still available. However, superimposed input pulses have been explicitly avoided. The pulses series are processed by the transmission chain model and the selection of the noise model has been essentially driven by the sake of simplicity. Thus, an 18


additive White Gaussian Noise (WGN) model has been chosen and the addition stage has been placed right before the ADC device, as shown in Fig. 2.9. We evaluated by simulation the behaviour of several digital filters (using Matlab, Simulink [1]). In particular, we focused on: • standard LP III order Butterworth filter; • adaptive LMS filter. Aiming at the best noise reduction, we compared the output of every digital filter to the digitalized output of the analog shaper. The results are presented in the following Section.

Figure 2.9: Simulated transmission chain.

This Simulation Scheme has been implemented both in Matlab and in Simulink. In Figs. 2.10, 2.11, 2.12 Simulink schematics, analog and digital chains are presented.

19


Figure 2.10: Simulink-implemented transmission chain schematics.

Figure 2.11: Simulink-implemented schematics: Analog Chain (green block in Fig. 2.10).

Figure 2.12: Simulink-implemented schematics: DGT Chain (blue block in Fig. 2.10).

20


2.3

Digital conversion and filtering algorithms

Let us consider the detection of two charged particles. The detector will produce two short current pulses with the amplitude proportional to the charge deposited by every single particle. In our simulation the two amplitudes are different, already transformed in voltage and expressed in normalized voltage unit (nvu). The first pulse is equal to 1 and the second to 0.5 nvu; the pulse duration is 0.5 ns, while the interarrival time is 0.7 µs. The bandwidth used to digitally represent the analog section is 5 GHz, while the digital bandwidth after the ADC is reduced to 50 MHz. The impulse response of the analog blocks simulated with Matlab is summed up in Fig. 2.13, while in Fig. 2.14 the same result elaborated with Simulink is reported. Adding to the shaper output a WGN signal one-sixth less powerful at the same sampling rate, we obtain a simulation of a noisy analog measurement (Fig. 2.15).

1

input preampl P/Z comp shaper

SHAPER

0.9 0.8

amplitude [nvu]

0.7

PREAMPLIFIER INPUT

0.6 POLE / ZERO COMP 0.5 0.4 0.3 0.2 0.1 0

0

500

1000 time [ns]

1500

2000

Figure 2.13: Analog outputs obtained with Matlab simulation.

Putting our attention on the analog shaper output, Fig. 2.15a represents the noisy measurement from which we would like to extract the desired signal (Fig. 21


Figure 2.14: Analog outputs obtained with Simulink simulation: input(red), pre-ampl(cian), PZcomp(purple), shaper(yellow).

2.15b), after sampling and quantization. Note that the sampling operation is a simple down-sampling in our simulation. Quantization is performed by the model of ADC introduced in Sect. 2.2. Performing the Discrete Fourier Transform (DFT) of the digitized analog shaper output, we have its representation in the frequency domain. Its square modulus provides the Power Spectral Density (PSD) of the considered signal; it is a useful mathematical tool that shows the most important frequency components in the signal spectrum. In Fig. 2.16 we can see the PSD of the analog shaper desired output, where the signal power has been normalized to the unit. The most significant frequencies are bounded in the lower part of the spectrum; as a consequence, the starting point of our analysis is a LP digital filter. Furthermore, Shannon Theorem is satisfied because the band involved is 20 MHz and we sample this signal with a sampling frequency of 100 MHz (≥ 2×20 MHz). The Matlab simulated noisy and desired digital signals are compared in Fig. 2.17.

22


NOISY analog signal 1.5 amplitude [nvu]

a 1 0.5 0 −0.5

0

200

400

600

800

1000 1200 time [ns]

1400

1600

1800

2000

DESIRED analog signal 1.5 amplitude [nvu]

b 1 0.5 0 −0.5

0

200

400

600

800

1000 1200 time [ns]

1400

1600

1800

2000

Figure 2.15: Continuous shaper output: noisy(a) vs desired(b).

0.3

0.25

PSD

0.2

0.15

0.1

Band of 20 MHz

0.05

0

5

10

15 20 frequency [MHz]

25

Figure 2.16: PSD of analog shaper output.

23

30


Noisy vs desired DGT signals 1

noisy desired

amplitude [nvu]

0.8

0.6

0.4

0.2

0

−0.2 0

500

1000 time [ns]

1500

2000

Figure 2.17: Matlab simulated noisy and desired digital signals.

24


2.3.1

Butterworth LP digital filter

The well known Bessel, Butterworth and Chebyshev standard filter families, with transfer functions from the II to the V order, have been compared in terms of SNR and PD estimations. The results are reported in Tabs. 2.1 and 2.2. In this Section, the SNR is calculated as the ratio between the desired signal power and the noise power, expressed in dB. The PD, or percentage error, is defined here as the relative difference between the desired signal peak and the filtered one, expressed in percentage. In case of several peaks, we decided to focus on the case with the highest distortion. Order Butterworth Bessel Chebyshev

II III 5.50 5.76 4.22 3.52 2.88 4.29

IV V 5.88 5.95 3.05 2.71 4.95 5.37

Table 2.1: SNR improvement [dB], our choice in bold typeface.

Order Butterworth Bessel Chebyshev

II III 7.72 8.34 5.89 7.19 7.45 9.60

IV V 10.45 11.40 7.93 8.38 12.66 12.58

Table 2.2: PD [% error], our choice in bold typeface.

The percentages shown in Tab. 2.2 all refer to the same worse condition. From these results the best performances both in noise reduction and in peak reproducibility are obtained with the Butterworth family. The highest noise reduction is obtained with the III order transfer function, while the highest peak reproducibility (lowest percentage error) is obtained with the II order. We chose the III order because our aim was the noise reduction and we set the cut-off frequency to 20 MHz, i.e., to the bandwidth of the input signal. The normalized transfer function in the continuous complex frequency domain is: H(s) =

s3

+

2s2

1 + 2s + 1

(2.1)

Denormalizing this transfer function to the cut-off frequency (Figs. 2.18, 2.19), we obtain: 25


Figure 2.18: Butterworth III order denormalized analog transfer function, 3D plot (right) and projection on complex plain (left).

Figure 2.19: Butterworth III order denormalized analog transfer function, 2D project mask obtained from the projection of the 3D plot on the plain real positive freq. vs modulus.

26


H(s) =

1.98 · 1024 s3 + 2.51 · 108 s2 + 3.16 · 1016 s + 1.98 · 1024

(2.2)

0.075 + 0.226z −1 + 0.226z −2 + 0.075z −3 1 − 0.827z −1 + 0.515z −2 − 0.087z −3

(2.3)

To convert this analog transfer function in the digital domain, we used a digital transfer function H(z) obtained from bilinear transform of H(s) [13] with a cut-off frequency according to the bandwidth of the desired input signal(20 MHz): H(z) =

Once evaluated, the transfer function coefficients are fixed and time independent by definition of standard filter. The typical structures to implement a digital IIR filter corresponding to H(z) will be presented in Sect. 2.5.

2.3.2

MMSE noise canceller

The standard III order LP Butterworth analog filter has fixed parameters (transfer function coefficients) that are known and calculated with Butterworth III order polynomials [13], denormalized at a particular cut-off frequency, bilinear transformed and then implemented in the digital domain. However, they do not depend on the characteristics of the specific considered signal. In this Section, we introduce a filtering algorithm whose parameters are dynamically calculated and adapted to the input signal in real time, the LMS algorithm (see [14] for a complete development). We are interested in the Minimum Mean Square Error (MMSE) noise canceller implementation of this filter and in its Finite Impulse Response (FIR) form. If a process d(n) is to be estimated from an observed process x(n) corrupted by the noise v(n): x(n) = d(n) + v(n)

(2.4)

and if we do not have any kind of information about d(n) or v(n), it is not possible to separate the signal from the noise. However, given a reference signal, this problem can be solved [14]. In the nuclear detector applications, here considered, we cannot have a reference signal and so we can adopt a different approach. We simply delay the process x(n) of n0 samples (Fig. 2.20), where x(n) is the measurement, d(n) b is the desired component, d(n) is an estimate of d(n), v(n) is the noise component uncorrelated from d(n), b v (n) is an estimate of v(n). The typical structures to implement a digital LMS FIR filter will be introduced in Sect. 2.5. In our model we can assume that d(n) is a narrowband process (Fig. 2.16) and that v(n) is a broadband process with: E {v(n) · v(n − k)} = 0, |k| ≥ k0 27

(2.5)


Figure 2.20: MMSE noise canceller using a LMS filter.

where E{·} is the statistical expectation and, if v(n) is white, then k0 = 1. So shifting the reference of at least k0 samples, the noise component of the signal x(n − n0 ) is uncorrelated with the noise of the measured signal x(n). Therefore, if k0 ≤ n0 ≤ k1 , the delayed process x(n − n0 ) will be uncorrelated with the noise v(n), but correlated with d(n) (from the condition n0 ≤ k1 ). Thus, the samples of x(n − n0 ) may be used as a reference signal to estimate d(n) as illustrated in Fig. 2.20. The problem we want to solve is how to obtain an estimate of the current sample d(n) of the desired signal from a set of M = k1 − k0 + 1 previous samples of the measured signal: xM (n − n0 ) = [x(n − n0 ),x(n − n0 − 1), . . . ,x(n − n0 − M + 1)]T .

(2.6)

To do this, the observation vector xM (n − n0 ) must be filtered by a proper linear predictor, designed as a FIR filter with coefficients: w M = [w0 ,w1 , . . . ,wM −1 ]T

(2.7)

b so that the filtered signal, d(n), is written as: b = wH · xM (n − n0 ) = d(n) M

M −1 X k=0

wk∗ · x(n − n0 − k)

(2.8)

H where wM is the Hermitian transpose of wM . The optimum design of the filter coefficients can be made through the minimization of the Mean Square Estimation Error (MSE) [13], defined as:

E |e(n)|2 28

(2.9)


b where e(n) = d(n) − d(n) is the estimator error. It is a known result of the MMSE filter design theory [14], [15] that the optimum set of filter coefficients for the linear prediction problem, stated as before, is given by: Rxx · wM = r xd

(2.10)

where Rxx = E{xM (n − n0 · xH M (n − n0 )} is the M × M autocorrelation matrix H of the input process, xM is the Hermitian transpose of xM , and r xd = E{xM (n − n0 ) · x(n)} is the cross-correlation vector between the past observation xM (n − n0 ) and the current one x(n), which contains the desired component d(n). However, in the case of the considered experiments, the nonstationarity of the observed process suggests to choose an iterative, adaptive implementation of the above formulation, known as LMS adaptive filter [14], [15]. Using a one-point sample mean (for a more complete discussion of the LMS algorithm the reader is referred to [15]), the update equation assumes a simple form known as the LMS Algorithm: wM (n + 1) = w M (n) + µ · e(n) · x∗M (n − n0 )

(2.11)

where w M (n + 1) is a new vector of filter coefficients at time n + 1, wM (n) is the filter coefficients vector at time n, e(n) is the error at time n, x∗M (n − n0 ) is the complex conjugate of the measurement at time n − n0 , n0 is the introduced delay, and µ is the stepsize. It is a positive number that affects the rate at which the weight vector w M (n) moves down towards a stable solution. Since this Section is preliminary for the implementation of these filtering algorithms on FPGAs, we need to take into account their computational complexity. Let us consider the LMS complexity in terms of additions and multiplications: Eq. 2.11 requires one addition to compute the error e(n) and one multiplication to form the product µe(n), M multiplications, and M additions to update the filter coefficients. Finally, M multiplications and M −1 additions are necessary to calculate the b output, y(n) = d(n), of the adaptive filter. Thus, a total of 2M + 1 multiplications and 2M additions per output point are required. The choice of the stepsize µ corresponds to a tradeoff among: • the SNR, in order to have a rough estimation of the real noise reduction; • the time dependence of the squared error function (e2 (n)). The e(n) function is a positive or negative quantity involved in Eq. 2.11 responsible for real time correction of the filter coefficients. If the algorithm converges to a stable set of coefficients the correction, as a function of time, and its squared estimate, should have a decreasing behaviour; • the coefficient settlement during the measurement or the simulation in order to understand if the algorithm has reached a stable solution. 29


2.4

Behaviour comparison of simulated results

In Fig. 2.17 the desired digital signal is plotted superimposed on the noisy digital signal. The filtered signal obtained with an IIR Standard Butterworth LP III order digital filter is presented in Fig. 2.21. The input SNR is 8.41 dB. Using the Butterworth filter this quantity rises to 14.17 dB with an improvement of 5.76 dB.

1

amplitude [nvu]

0.8 noisy desired Butt. III

0.6

0.4

0.2

0

−0.2 0

500

1000 time [ns]

1500

2000

Figure 2.21: Butterworth III order filtered signal vs. noisy and desired signals.

However, this filter introduces a peak amplitude distortion that is worse for the first of the two processed pulses. In Fig. 2.21, the amplitude of the first peak for the Butterworth filtered signal is greater than the desired signal, with a distortion of 8.34 %. The FIR LMS order n was fixed to 4 to introduce a medium level of complexity with a reasonable elaboration time. Using a LMS filter the performances are a function of the stepsize µ, as shown in Tab. 2.3 and Figs. 2.22 - 2.24. A quick consideration must be done for Fig. 2.24: it represents a functional to be optimized to find the best filter performances. Since we want to enhance the SNR and reduce the PD at the same time, multiplying the SNR by the inverse of the PD we obtain a quantity to be maximized, it is the result presented in the last aforementioned Figure. The percentage error is calculated as for the Butterworth filter for the first peak, that is for the highest distortion condition. 30


LMS

Butterworth

stepsize µ error [%] 0.05 36.88 0.10 15.54 0.15 5.70 0.18 0.06 0.20 2.63 0.25 5.93 0.30 6.67 8.34

SNR improvement [dB] 5.51 6.35 6.58 6.57 6.52 6.33 6.09 5.76

Table 2.3: LMS performances vs µ, our choice in bold typeface.

7

6.5

SNR [dB]

6

5.5 LMS MAX: µ =0.16 −− SNR =6.59 dB Butterw. SNR: 5.76 dB

5

4.5

4

0

0.1

0.2

stepsize µ

0.3

0.4

0.5

Figure 2.22: SNR expressed in dB as a function of stepsize µ.

31


90 LMS MIN: µ =0.18 −− PD =0.06 % Butterw. PD: 8.34%

80

Peak Distortion [%]

70 60 50 40 30 20 10 0

0

0.1

0.2

stepsize µ

0.3

0.4

0.5

Figure 2.23: PD expressed in % as a function of stepsize µ.

12000

10000

arbitrary unit

8000 curve max = (0.18, 11840) SNR = 6.57 dB PD = 0.06 %

6000

4000

2000

0

0

0.1

0.2

µ

0.3

0.4

0.5

Figure 2.24: SNR / PD functional to be maximized for the optimal solution retrieval.

32


In Fig. 2.25, the LMS filtered signal with µ = 0.18 is presented; the output SNR is 14.98 dB, so the enhancement is of 6.57 dB, and the PD is also enhanced (lower percentage error). Indeed, in Fig. 2.25 the amplitude of the first peak for the LMS filter nearly matches the desired signal, the distortion being only of 0.06 %.

1

amplitude [nvu]

0.8 noisy desired LMS

0.6

0.4

0.2

0

−0.2 0

500

1000 time [ns]

1500

2000

Figure 2.25: LMS filtered signal (µ = 0.18) vs. noisy and desired signals.

The choice of the stepsize corresponds to a tradeoff among the evaluated SNR enhancement, squared-error function decreasing behaviour (Fig. 2.26), and coefficients settlement (Fig. 2.27) introduced in the previous Section.

33


0.35 LMS Sq. Error 0.3

amplitude [nvu]

0.25

0.2

0.15

0.1

0.05

0

0

500

1000 time [ns]

1500

2000

Figure 2.26: Squared error amplitude vs. time: decreasing behaviour

LMS w2 0.6

0.4

0.4

coeff. value

coeff. value

LMS w1 0.6

0.2 0

0

500

1000 1500 time [ns] LMS w3

0 −0.2

2000

0.8

0.3

0.6

0.2

coeff. value

coeff. value

−0.2

0.2

0.4 0.2 0

0

500

1000 1500 time [ns]

2000

0

500

1000 1500 time [ns] LMS w4

2000

0

500

1000 1500 time [ns]

2000

0.1 0 −0.1

Figure 2.27: LMS four coefficients (µ=0.18) behaviour: a stable solution is reached.

34


2.5

FPGA-oriented implementations

After proper design and simulation, in order to perform digital signal processing on real-time signals, filtering must be implemented on an ASIC (Application Specific Integrated Circuit) or on a programmable board, as a FPGA (Field Programmable Gate Array). A Xilinx Virtex 4 ML402 FPGA [2] will be considered in this Section. For this purpose, filter structures suitable for hardware implementation must be developed and optimized. The Matlab and Simulink simulations of Butterworth III order LP and adaptive LMS filtering algorithms has been discussed in the previous Section. However, the direct translation of those structures in VHDL leads to board consumptions too high for an FPGA implementation. For this reason, some changes have been introduced and in this Section they will be discussed and compared with the original simulated schematics.

2.5.1

Butterworth LP digital filter

For sake of simplicity the digital Butterworth III order transfer function is reported here:

H(z) =

0.075 + 0.226z −1 + 0.226z −2 + 0.075z −3 1 − 0.827z −1 + 0.515z −2 − 0.087z −3

(2.12)

It is important to remember that, once evaluated, the transfer function coefficients are fixed and time independent by definition of standard filter. In our simulations H(z) was implemented with a Direct Form (DF) II structure, as shown in Fig. 2.28. The numerator coefficients are the multipliers of the rightern subchain, the denominator ones of the leftern. This second half is fed back and its contribution is added to the input signal. Translating this structure in VHDL language for the implementation on the Virtex 4, signals coming from the lowest register, passing into the fed back (leftern) subchain, encounter 2 multipliers and 4 adders (Fig. 2.28). This path is the so-called combinatorial path and represents the distance that a signal has to cover between two consecutive registers, or between one register and the filter output [6]. Long combinatorial paths take more time to execute, so they limit the maximum compile rate of the FPGA. The maximum working frequency is the inverse of this time. As an example, in the case of the Butterworth III order filter, the implementation with a DF II structure leads to a combinatorial path that imposes a maximum working frequency of 34 MHz. Since this rate could not be enough to cope with the clock rate of some detectors of a nuclear experiment (as expected for Panda [3]), 35


the filtering structures must be optimized to increase the FPGA maximum working frequency. To improve this feature we have to break the longest combinatorial path. This was done by moving to a DF I transposed structure for translating the same transfer function (Fig. 2.29). In this case the maximum combinatorial path is composed of only 1 multiplier and 2 adders (Fig. 2.29), thus the maximum working frequency rises up to 63 MHz.

36


Figure 2.28: IIR Butter. III ord. filter with DF II structure, Simulink schematics. Longest combinatorial path highlighted in dark green.

Figure 2.29: IIR Butter. III ord. filter with DF I transposed structure, Simulink schematics. Longest combinatorial path highlighted in dark green.

37


2.5.2

MMSE noise canceller

The structure used for the LMS simulation, translated in VHDL, presents a very long maximum combinatorial path (3 multipliers and 5 adders) leading to a maximum working frequency of 22 MHz (LMS1, Fig. 2.30). Also in this case, it is possible to optimize the hardware structure and enhance this rate. In order to break the longest combinatorial path, some extra registers were added. The optimized adaptive filter structure (LMS2) is shown in Fig. 2.31. With these changes in the structure, the maximum combinatorial path starts after the register under the first coefficient subchain, passes through 4 adders and the multiplier with the stepsize µ as second input. This path is shorter thus the maximum working frequency increases to 58 MHz. Since the LMS filter structure was changed, we found, as expected, a different value for the stepsize µ that optimizes our requirements on SNR maximization and PD reduction. It is worth noting that for both LMS1 and LMS2 structures every adaptive coefficient is implemented with a dedicated subchain, fed by the multiplication of the stepsize µ with the difference between the noisy signal and the adaptive filter output (Figs. 2.30, 2.31).

38


Figure 2.30: FIR LMS 4-coefficients structure Matlab simulated (LMS1), Simulink schematics. The longest combinatorial path is highlighted in dark green, mu is the stepsize.

Figure 2.31: FIR LMS 4-coefficients structure implemented for FPGA board (LMS2), Simulink schematics. The longest combinatorial path is highlighted in dark green, mu is the stepsize, the added registers are highlighted in light green.

39


2.6

Discussion of FPGA-oriented results

In this Section the performance of the FPGA digital filtering implementation is discussed with reference to nuclear detector requirements. For the sake of comparison with the simulation results (see Sect. 2.2 and [10]), the same noisy signal has been used for the FPGA filtering. The information pulses are modelled with a series of successive finite support waveforms, with very narrow time duration (0.5 ns) and random times of arrival. No specific assumptions had been made about the mathematical model of the times of arrival, since it was intended to perform a general analysis not referred to a specific detector. White Gaussian Noise (WGN) had been added to the desired signal.

2.6.1

Filter performance

In Fig. 2.32 and in Tab. 2.4 the most relevant simulation results are presented. As reported, the LMS filter introduces a much lower PD and a slightly higher SNR than Butterworth III order filter. The PD is evaluated for the filtered peak featuring the highest distortion (the first in this analysis) (see Sect. 2.2 and [10]).

1

amplitude [nvu]

0.8

0.6 Noisy Desired Butt III LMS1

0.4

0.2

0

−0.2 0

500

1000 time [ns]

1500

2000

Figure 2.32: Matlab simulated filter comparison.

The direct VHDL implementation of these structures leads to the same values of 40


Filter type Butt. III DF II LMS1

Stepsize µ 0.18

PD [%] 8.34 0.06

SNR improvement [dB] 5.76 6.57

Table 2.4: Performance of Matlab-simulted Butterworth and LMS filters.

SNR and PD of Matlab simulated filtering algorithms. The plot of the filtered signals is presented in Fig. 2.33. In the VHDL translation both Butterworth and LMS filtered signals present a delay of 2 sampling periods with respect to the Matlab simulated ones. Indeed, in every structure of the previous Section the two registers for input and output storage are not shown. This corresponds to a usual FPGA implementing rule chosen to avoid a wrong numerical representation of signals when high working frequencies are involved.

1

amplitude [nvu]

0.8


0.4

0.2

0

−0.2 0

500

1000 time [ns]

1500

2000

Figure 2.33: Matlab simulated structures implemented on FPGA.

As pointed out in the previous Section, a direct VHDL translation of the Butterworth DF II structure (Fig. 2.29) led to a low maximum working frequency. Thus, we adopted the translation of the DF I transposed structure (Fig. 2.28). The processing and the performances of PD and SNR do not change for the Butterworth filter because the same transfer function has been translated. A different 41


consideration has to be made for the VHDL translation of the LMS filter. Since the LMS1 structure worked at low frequency, some registers were added to break the too long combinatorial path. This insertion changed the structure (LMS2) giving higher working frequency. Therefore, the LMS2 structure leads to a different best stepsize value, but also to a higher PD and a lower SNR than the Matlab simulated LMS1 structure (Tab. 2.4). This is the price that must be paid to work at a frequency two times and a half higher. Anyway, LMS filter continues to match better the requirements related to the energy resolution and detector efficiency than the Butterworth one also in the implemented FPGA form (Tab. 2.5, Figs. 2.34 2.36). The functional to be maximized has been calculated as reported in Sect. 2.4. In Fig. 2.37 the optimized Butterworth (DF I transposed) and LMS (LMS2) filter outputs are shown. The LMS2 filter shows a delay of 8 sampling periods, with respect to the LMS1 structure (Fig. 2.33), due to the added registers that shorten the combinatorial path. The Butterworth FPGA filtered signal has no delay, because no register is added. Filter type Butt. III DF I LMS2

Stepsize µ PD [%] 8.35 0.27 0.49

SNR improvement [dB] 5.76 5.91

Table 2.5: Performance of FPGA-optimized Butterworth and LMS filtering structure.

42


6.5

6

SNR [dB]

5.5

5

4.5

4 LMS MAX: µ =0.39 −− SNR =6.22 dB Butterw. SNR: 5.76 dB

3.5

3

0

0.1

0.2

stepsize µ

0.3

0.4

0.5

Figure 2.34: SNR expressed in dB as a function of stepsize µ.

100 LMS MIN: µ =0.27 −− PD =0.49 % Butterw. PD: 8.35%

90 80

Peak Distortion [%]

70 60 50 40 30 20 10 0

0

0.1

0.2

stepsize µ

0.3

0.4

Figure 2.35: PD expressed in % as a function of stepsize µ.

43

0.5


900 curve max = (0.27, 886) SNR = 5.91 dB PD = 0.49 %

800 700

arbitrary unit

600 500 400 300 200 100 0

0

0.1

0.2

µ

0.3

0.4

0.5

Figure 2.36: SNR / PD functional to be maximized for the optimal FPGA-oriented filtering solution retrieval.

FPGA Filtering Comparison 1

amplitude [nvu]

0.8


0.4

0.2

0

−0.2 0

500

1000 time [ns]

1500

Figure 2.37: FPGA-optimized filtering structures output.

44

2000


2.6.2

Power consumption

In the implementation of the filtering structure on FPGA, it is very important to evaluate the consumption of the board internal components (i.e. the percentage of utilized memory slices and digital signal processors, the number of needed logic levels). This analysis is indeed critical if we consider the total number of boards that should be used in a nuclear particle experiment. We have compared the different filtering structures described in the previous Section also under this point of view. As reported in Tabs. 2.6 and 2.7, the DF I transposed and the LMS2 structures optimized for FPGA reach not only higher maximum working frequency, but also need less components for their implementation than the simulated and directly VHDL translated filtering structures (DF II and LMS1). Butterworth DF II DF I transp.

# Memory slices (%)

Digital Signal Processing applied to Physical Signals

Digital Signal Processing applied to Physical Signals

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